Estimating total mass inside the Sun's orbit

Brad 05/13/2018. 1 answers, 32 views
homework-and-exercises dark-matter galaxies milky-way galaxy-rotation-curve

I'm trying to estimate the local density of dark matter at a given radius, $r=R_0=8\text{kpc}$. I also have values for the core radius, $a=5\text{kpc}$, and circular velocity of the Sun, $v_{\text{circ}}=220\text{kms}^{-1}$. I start with the dark matter density profile


from which I can obtain a relation for the mass interior to the radius r


$\rho_0=M(r)/(4\pi a^2(r-a\text{tan}^{-1}(r/a)+const))$

To get the local dark matter density at the given radius I assume I need to substitute the above relation into the first equation.

However, I'm also told that the dark halo contributes half of the total mass inside the Sun's orbit but I don't know how to calculate the mass interior to the Sun (probably missing something very simple here). Any help is greatly appreciated.

1 Answers

Brad 05/14/2018.

The density profile given in the question is for a spherical halo, meaning that we can assume the distribution of mass in the halo to be spherically symmetric - allowing us to use the spherically symmetric gravitational acceleration


and the centripetal acceleration for a circular orbit


Equating these and rearranging we obtain


which can be simply substituted in to the equation for $\rho_0$, which can then be substituted into the density profile given. Using the values given above I get:

$\rho(R_0)=1.85493\times10^{-21}\text{kgm}^{-3}\\\ =0.02726M_{sun}\text{pc}^{-3}$

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